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Lesson starter: Guess the output

A table of values is drawn on the board with three completed rows of data.

• Additional values are placed in the input column. What output values should be in the output column?

• After adding output values, decide which rule fits (models) the values in the table and check that it works for each input and output pair. Four sample tables are listed below.

Key Ideas

■ A rule shows the relation between two varying quantities. For example: output = input + 3 is a rule connecting the two quantities input and output The values of the input and the output can vary, but we know from the rule that the value of the output will always be 3 more than the value of the input. Rules can be thought of as function machines:

+ 3 input output

■ A table of values can be created from any given rule. To complete a table of values, we start with the first input number and use the rule to calculate the corresponding output number. we do this for each of the input numbers in the table. For example: For the rule output = input + 3, If input = 4, then output = 4 + 3 = 7

• Replacing the input with a number is known as substitution input 1 2 3 4 5 6 output 6 7 8 9 10 11

■ Often, a rule can be determined from a table of values. On close inspection of the values, a relationship may be observed. Each of the four operations should be considered when looking for a connection.

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Building Understanding

1 State whether each of the following statements is true or false. a If output = input × 2, then when input = 7, output = 14.

b If output = input 2, then when input = 5, output = 7.

c If output = input + 2, then when input = 0, output = 2.

d If output = input ÷ 2, then when input = 20, output = 10.

2 Choose the rule (A to D) for each table of values (a to d).

= input 5

Example 21 Completing a table of values

Copy and complete each table for the given rule.

= (3 × input) + 1

Now you try

Copy and complete each table for the given rule.

each input value in turn into the rule.

Example 22 Finding a rule from a tables of values

Find the rule for each of these tables of values.

Now you try

Find rule for each of these tables of values.

Exercise 2J

Copy and complete each table for the given rule.

Copy and complete each table for the given rule.

4 State the rule for each of these tables of values.

5 6 7 8

6 7 8 9

5 State the rule for each of these tables of values.

8 3 1 14 a Write the rule above using input and output, rather than radius and diameter. b Copy and complete the missing values in the table.

6 A rule relating the radius and diameter of a circle is diameter = 2 × radius. In the following table, radius is the input and diameter is the output.

2 3 7

7 Find a simple rule for the following tables and then copy and complete the tables.

8 Copy and complete each table for the given rule.

9 Another way of writing output = input + 3 is to state that y = x + 3. Write the rule using y and x for the following.

10 It is known that for an input value of 3 the output value is 7 a State two different rules that work for these values. b How many different rules are possible? Explain.

11 Many function machines can be ‘reversed’, allowing you to get from an output back to an input. For instance, the reverse of ‘+ 3’ is ‘ 3’ because if you put a value through both machines you get this value back again, as shown: a output = input + 7 b output = input 4 c output = 4 × input d output = 2 × input + 1 e output = input ÷ 2 + 4

The reverse rule for output = input + 3 is output = input 3 Write the reverse rule for the following.

ENRICHMENT: Finding harder rules – – 12 b Write three of your own two-operation rules and produce a table of values for each rule. c Swap your tables of values with those of a classmate and attempt to find one another’s rules.

12 a The following rules all involve two operations. Find the rule for each of these tables of values.